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Design and Incentives for Cooperative Solutions 
to Water Problems in the San Joaquin Valley 


under contract for the 


San Joaquin Valley 

Drainage Program 

September 1990 


This report presents the results of a study conducted for 
the Federal -State Interagency San Joaquin Valley Drainage 
Program. The purpose of the report is to provide the Drainage 
Program agencies with information for consideration in developing 
alternatives for agricultural drainage-water management. 
Publication of any findings or recommendations in this report 
should not be construed as representing the concurrence of the 
Program agencies. Also, mention of trade names or commercial 
products does not constitute agency endorsement or 

The San Joaquin Valley Drainage Program was established in mid- 1984 as a 
cooperative effort of the U.S. Bureau of Reclamation, U.S. Fish and Wildlife 
Service, U.S. Geological Survey, California Department of Fish and Game, and 
California Department of Water Resources. The purposes of the Program are to 
investigate the problems associated with the drainage of irrigated 
agricultural lands in the San Joaquin Valley and to formulate, evaluate, and 
recommend alternatives for the immediate and long-term management of those 
problems. Consistent with these purposes. Program objectives address the 
following key areas: (I) Public health, (2) surface- and ground-water 
resources, (3) agricultural productivity, and (4) fish and wildlife resources. 

Inquiries concerning the San Joaquin Valley Drainage Program may be 
directed to: 

San Joaquin Valley Drainage Program 
2800 Cottage Way, Room W-2143 
Sacramento, California 95825-1898 

Design auid Incentives for Cooperative Solutions 
to Water Problems in the San Joaquin Valley 


Edna Loehcunn 

Purdue University 

West Lafayette, Indiana 47907 

under U.S. Bureau of Reclamation 

contract O-PG-20-04140 


Ariel Dinar 

University of California, Riverside 

Riverside, California 92521 

under U.S. Bureau of Reclamation 

contract 9-PG-20-03380 

September 1990 

Design and Incentives for Cooperative Solutions 
to Water Problems in the San Joaquin Valley 

Executive Summary 

This report presents a method for the design of economic 
incentives leading to regional management of water problems 
relating to conservation and drainage water quality in the 
West side of the San Joaquin Valley. A game theory paradigm 
is the basis for design of these incentives. Voluntary 
adoption of improved water use technologies is the solution 
of a cooperative game among producers, consumers, and a 
regional manager. The method described results in solutions 
which are both economically efficient and consistent with 
political considerations. 

Table of Content 


Introduction 1 

Game theory concepts applied 

to the water use problem 3 

Payoff functions for game players 7 

The noncooperative solution 11 

The cooperative solution 15 

Core Conditions 17 

Conclusions 19 

References 20 

Appendix A 23 

Appendix B 27 



Figure 1 : A cooperative solution and 

the corresponding NNE "threat point" 4 

Figure 2: Water use, noncooperative 

solution compared to the status quo 14 

Figure 3: Illustration of the core with 

three players 24 

Design of Incentives for Cooperative Solutions to 
Water Problems in the San Joaquin Valley 


Major portions of agricultural lands in the San Joaquin Valley have 
drainage and salinity problems. Irrigation water used for agricultural 
production may, under certain conditions, be associated with long and short 
term environmental pollution in terms of elevated levels of selenium and 
other trace elements in soil and water (SJVDP, 1989). The resulting 
externality problems derive from water use decisions made by individual 
producers in the San Joaquin Valley. 

Selenium contamination of wetlands and water supplies results from 
leaching. Selenium reduces the reproduction rates of fish (Saiki et al. 
1990) and waterfowl (Skorupa and Ohelendorf, 1990), thereby reducing 
benefits to those engaged in recreation such as hunting, fishing, bird- 
watching (Loomis et al , 1990). For high enough pollution levels, health 
effects may also occur to those who consume local water (Klassing, 1990). 
Food consumption is not considered to be a problem for most consumers 
because food is purchased from multiple sources. 

Because of the topography and the dominant direction of groundwater 
movement, a high water table resulting from irrigation by upslope producers 
may cause damage to crops of downslope producers. Leaching of salts due to 
a high water table may also have negative effects on agricultural production 

both upslope and downslope (Rhoades and Dinar, 1990). 

1. In the land area affected significantly by drainage problems, on a 
regional basis, the dominant direction of groundwater movement is vertical, 
reflecting the influence of decades of heavy groundwater pumping. Near- 
surface (lateral flows) movement may occur on the farm sacle, especially in 
the presence of relict upslope/downslope movements are important in 
individual farming operations but may be negligible for regional hydrologic 
patterns (Gilliom, 1989). 

Reducing irrigation water would be desirable not only to reduce 
pollution levels but also to improve efficiency of water utilization under 
conditions of water scarcity. Adoption of improved irrigation technologies 
is one means to alleviate at least some of these problems by reducing 
drainage (Rhoades and Dinar, 1990). Such technologies include use of new 
methods -- such as sprinkler and drip irrigation -- and use of improved 
management practices for traditional technologies -- such as reduced furrow 
length and timing of application. 

Regional systems for drainage, water treatment, and re-use of treated 
water could complement privately applied technologies. Because of economies 
of scale, improved drainage water quality may be achieved at lower joint 
cost by the combination of improved individually applied technologies and 
regional systems. 

A regional cooperative water system for improving water quantity and 
quality control has the nature of a public good in that it would provide 
benefits jointly to producers and consumers who would value such a system 
differently. In spite of potential benefit from cooperation, as with other 
types of public goods, improvements in private technologies and regional 
systems may not be adopted unless there are incentives to do so. 

Cost sharing arrangements, through which joint costs are shared among 
agricultural producers and consumers of recreation and other goods related 
to water quality, may provide some incentives for cooperation. However, 
cost sharing by itself may not provide sufficient incentives for adoption of 
improved private technologies and regional management systems. Additional 
requirements that producers meet water quality objectives may need to be 
imposed by a regional manager as in Kilgour (1988). 

This paper presents a method for the design of economic incentives 
leading to a regional solution to the problems of water conservation and 
drainage water quality in the San Joaquin Valley. A game theory paradigm is 
the basis for design of these incentives. Game players include upslope 
producers, downslope producers, and consumers of recreation. The role of 
the regional manager in this game is to propose and enforce rules for the 
game consistent both with political power of the players and economic 
efficiency. Voluntary adoption of improved water use technologies is the 
desired solution of this cooperative game. 

Game Theory Concepts Applied to the Water Use Problem 
Game theory has been used to provide a basis for cost allocation for 
shared facilities (Young, 1985), viewing such situations as cooperative 
games. Less attention has been given to alleviating externality problems in 
a game setting and conclusions about the possiblity of cooperative solutions 
to externality games have been largely negative (Shapley and Shubik, 1969). 
Here we demonstrate that when an externality problem is combined with a 
public good which alleviates it, there may be a cooperative solution. 

In the game formulation for the externatility situation in the San 
Joaquin Valley, in order to provide incentives for cooperation, the 
game will include a "threat point" in addition to cost sharing rules. The 
threat point and cost sharing rules will be set by the regional manager to 
be consistent with political power and economic efficiency considerations. 

Economic efficiency is defined in terms of the production frontier. 
Figure 1 illustrates two production frontiers and four points important to 
defining the nature of the efficiency for this game. The interior 
production frontier represents efficient points in terms of agricultural 
production (F) and environmental quality (Q) when only private water 

technologies are used. An expanded frontier is obtained when improved 
private and regional water technologies are introduced because both more 
agricultural production and better drainage water quality may be produced. 
As discussed by Samuelson (1950) , only when the production frontier expands 
can all persons in society be assured of the possibility of being made 
better off. 


Figure 1. A Cooperative Solution and the Corresponding NNE "Threat Point" 

The status quo point (SQ) corresponds to a maximization of agricultural 
production with current private water use technologies. The point CP is the 
point of agricultural production and environmental quality most preferred by 
consumers . 

Parties damaged by externalities have incentives to organize 
politically to cause regional and/or state authorities to impose pollution 
and drainage standards, or equivalently to impose taxes on water use or 
pollutants. The countervening political power of producers limits the 
extent to which purely environmental objectives can be met. 

The noncooperative Nash solution (NNE) is a solution consistent with 
economic efficiency and political influence. It will be on the private 
production frontier between the status quo (SQ) and consumers' preferred 
solution (CP) . The relative influence of each player on the the resulting 

tradeoff between agricultural profit and drainage water quality is 
represented by the slope of the tangent line to the frontier. Therefore, 
there is a noncooperative Nash equilibrium solution corresponding to any set 
of political weights. 

The Nash equilibrium solution is not achieved in the case of 
externalities through private market decisions because of the absence of 
prices for environmental amenities or disamenities . The regional manager 
may cause the Nash solution to be achieved by setting taxes on water use 
and/or pollutants. 

The cooperative solution (CS) is also an efficient solution but it is 
preferred because it is on the expanded production frontier. Therefore, all 
parties may be made better off in the cooperative solution as compared to 
the NNE solution. However, the cooperative solution may not be achieved for 
reasons similar to the well-known public goods problem of welfare economics. 

The noncooperative Nash equilibrium, taxes on water use and land 
allocation used to achieve the NNE solution, can be used as a "threat point" 
for the cooperative game. The noncooperative solution can be imposed by the 
regional manager if the cooperative solution does not occur. 

In addition to the tax penalty for noncooperation , cost sharing for the 
regional system provides incentives for cooperation. If cost shares for all 
producers in the cooperative game are less than the taxes paid in the 
noncooperative case, and agricultural output is not diminished, then the 
cooperative solution should be preferred by producers to the noncooperative 
solution. Cost shares to cover costs of the regional facilities for the 
cooperative solution can be based on the same political weights used to 
define the noncooperative solution. 

Water scarcity can also be eased by regional water management because 
tail water can be collected and reused. Because of improved management. 

water use charges can be reduced, thus providing additional incentives for 

The core is the set of pareto optimal points for a game. All players 
better off at a point in the core compared to other outcomes. Therefore, 
core outcomes have potential for voluntary agreement. Figure 1 illustrates 
that two cases are possible. In the first case, because of the position of 
the production frontier, the core is points northeast of the NNE threat 
point. In the second case it also includes points to the northeast of the 
status quo. The second case leads to a better situation in terms of 
voluntary agreement because all can be made better off at a cooperative 
solution as compared to the status quo. Which situation will obtain will 
depend on both the noncooperative and cooperative solutions (as related to 
political weights) and the position of the expanded frontier in relation to 
the current frontier. 

The choice by producers and consumers between cooperative and 
noncooperative solutions to a game is related to a Prisoner's Dilemma 
situation. The Prisoner's Dilemma is a well-known problem in game theory 
literature which demonstrates that a noncooperative solution may be chosen 
over a cooperative solution even when the cooperative solution is pareto 
optimal. This situation may occur because there is no communication among 
players and there are no incentives to cooperate or penalties for 
noncooperation beyond the relative payoff values (Oppenheimer , 1990). By 
providing information exchange and additional incentives for cooperation, 
the presence of a regional manager may help to avoid the Prisoner's Dilemma. 

Below, we present these concepts more formally. A simplified model 
representing the situation in the San Joaquin Valley is used to illustrate 
how to define noncooperative and cooperative solutions and the core 
corresponding to political weights. 

Payoff Functions for Game Players 

To apply game theory and test whether a proposed cooperative solution 
does indeed lie in the core, preference or payoff functions must be defined 
for the players. Application of game theory to real world problems has been 
limited because preference functions are often not in common monetary units. 
Here, we will express consumers preferences (health and recreation concerns 
and expenditure for food) in terms of willingness to pay for improved 
drainage water quality, and preferences of producers in terms of profits, so 
that payoffs are in comparable monetary units. 

Below, these measures are defined more formally as related to 
environmental quality and alternative water technologies. 

Producer welfare will be described by profits . Per acre profit for 

each type of producer (upslope, n , and downslope w ) is revenue from food 
production minus variable costs of production, charges for water use (v per 
unit), taxes for water use and/or drainage, and annual costs per acre for 

water technologies used by each type of farmer, denoted by c(r ), c(t ) for 

technologies t and t . In the status quo case, taxes are zero. 

The upslope producer's yield (Y ) per unit land area is related to 

water use technology (t ) and water use (W ) per unit land area. 

Y^ - Y^(W^; r^). 
The upstream producer chooses water use, acres planted (subject to an 

acreage constraint A ), and water technologies to maximize profits. For the 
upstream producer, with a tax t on water use and a tax t. on land use, 
the total profit level achieved is defined by 

Max [pjY^ -(v+ t;;) W" -c(r^) - t;;] A^ 

u — u 
s.t. A < A 

y" - f(W^;r^). 
(To represent several crop activities, Y, W, and r can be vectors.) 

The downstream producers' yield is related to water use (W ) , as 
determined by the technology used, and the externality due to the water use 
of the upstream producer. Drainage caused by water use of upslope producers 
may reduce yield of downstream producers if there is excess water and 
salinity in the root zone. (On the other hand, water could be used more 
efficiently if drainage water from upslope could be captured and reused.) 
Since not all drainage water from upslope producers is received by downslope 
producers, a proportionality factor (k) adjusts for the amount of water 

Y^ > y'^(w'^, kw" A^ ; r^). 
The downstream producer chooses own water use, acres planted (subject to an 

acreage constraint A ), and technology to maximize profit. The profit for 
the downstream producer, with a tax t on water use and a tax t. on land, 
is : 

Max [p. Y^ -(V +t^) W'^ -c(r^) - t^] A^ 

d „d ^d ^ '^ ^ 

r , W ,A 

s.t. A^ < A^ 

Y^ - f(W^, kW^ A^; A. 

The total pollution load combines the effects of both upslope and 
downslope producers and depends on the land and water use decisions and 

technology used by each producer. The pollution (S ) produced by the 
upstream producer can be described by 

that is, pollution is proportional to the amount of water used and the 

number of acres irrigated and drained. The proportionality factor 5 
depends on the water technology chosen and the topography and soils 
upstream. The downslope producer's own effect on pollution is similarly 
represented, except that the resulting pollution includes both effects from 
his/her own water use and the drainage from the upstream producer: 

The total pollution effect (S) is the sum of S and S : 

s = S^ + S^. 
Note that it is proportional to total water used by each producer. (S may 
be a vector if there are several pollutants.) 

Consumer welfare is determined by income, pollution, and price of food 
and other goods. Consumer welfare can be represented by the indirect 
utility function: 

U - U(M, S, Pf-P^.Pr- P^) 

where M denotes initial wealth or income, S is the pollution level, and p^ 

denote respectively the prices of food, health, recreation, and other goods. 

It is assumed that pollution reduces consumer utility by affecting 
health and recreation. Thus, improvement of drainage quality would increase 
consumer welfare. Since utility is also affected by market prices for food, 


any food price increases due to cost of water improvements would have an 
offsetting negative effect on consumers. However, it is not required here 
that improvements in environmental quality be financed through increased 
food prices. We will assume that the market for products grown in the 
region is open so that food prices are not affected. 

Because utility is not in dollar units, it is not directly comparable 
to producer profits. The equivalent variation measure of willingness to pay 
provides a dollar measure of welfare which gives the same ranking of 
outcomes as utility. The amount of money (WTP) which is equivalent to a 
change in environmental quality satisfies the following relation: 

U(M - WTP, S? Pf.Ph.Pr- Pz^ " ^^"' ^' • PfPh'Pr- ^z^ 

when the pollution level is reduced from S to S ' , with S > S' . 

For example, for the linear expenditure system, the indirect utility 
function as related to pollution and prices is: 

^ ,_- , , , ;3z /9f i9h ^r 
Ip^ + r(S))7j.]/ P^ Pf Ph Pr 

U - [M - p^7^ - pf7f- (Ph+ h(S))y^ 

where 0f , 0h , 0r , fiz , denote attribute importance weights and 7. denote 

minimum acceptable levels respectively for food, health, recreation and 
other expenditures. Effects of increased contamination (S) cause 
expenditures needed to maintain these minimum levels to increase (i.e., 
h(S) , r(S) are positive and increase with S) . An improvement in 
environmental quality reduces the costs of purchasing the minimum level of 
recreation and health. In this case, willingness to pay is derived to be: 

WTP(S'; S°) = (h(S°) - h(S'))7j^ + (r(S°) - r(S')l7j. 

+ [M-p^72-Pf7f-(p^+h(S ))7^-(p^+r(S ))7^: 



^W p ^r(S')^^^- 



That is, willingness to pay for an improvement in environmental quality (a 
reduction in pollution) can be expressed as the reduced expenditure to 
purchase the minimum acceptable health and recreation levels plus a share of 
the overall expenditure. 

Below, willingness to pay (WTP(S; S )) will denote consumer welfare as 
a function of improved water quality levels (S) in comparison to the initial 

level S° . Note that 3WTP/3S <0 , i.e. as pollution decreases, willingness to 
pay increases. 

The Noncooperative Solution 
The noncooperative Nash equilibrium solution (NNE) lies along the 
current production frontier, when only individual, rather than regional, 
water technologies are used. The solution satisfies a tradeoff between 
drainage water quality and agricultural production defined in terms of 
political weights Iq). 

The noncooperative (NC) solution is an equilibrium such that producers 
individually choose water use and water technologies to maximize profits in 
response to taxes or standards. Taxes or standards on water use and 
contaminant levels are set by the regional manager. These taxes or 
standards induce producers to use water technologies to achieve a given NNE. 
Taxes reduce profit levels of producers compared to their profits at the 
status quo. As taxes on water use and/or pollution are increased to reflect 
a greater weight on water quality relative to profits, there are greater 
incentives for producers to adopt technologies which reduce water use and 
pollution but profits could also be lower. (Tax revenues by law would have 
to be returned to the system but would not necessarily be given back to 
producers directly.) 


According to a theorem of welfare economics (Negishi) , an equilibrium 
solution can be found by maximizing a weighted sum of payoff functions for 
game players. For the NNE solution, joint welfare is expressed as the 
weighted sum of payoff functions for consumers and producers; it is 
maximized over the set of private water technologies, acres planted, and 
water use for each producer. The optimal technologies determine the 
pollution load affecting consumers. Constraints for the joint maximum 

include total water availability (W) , yields, and pollution production 
functions . 

The joint welfare optimization problem corresponding to the 
noncooperative Nash equilibrium is: 

JW(q; NNE) - Max q WTP(S; S°)+ a [p^Y^ -c(r^) - vW^l a'^ 


+ Q^IPfY^ -c{t'^) -vW'^] A^ 

,u ^ tu 
s. t . A < A 


A^^ + A^ W^ < W 

„u ,u, u, ,,u ,u 
S - 5 (r ) W A 

d ^d, d, ,,,d ,d, , ,,u .u. 
S =(5(r)(w A + kW A) 

s - s" + S<^ 

y" - Y^CW^ir"") 

Y^ - y'^CW^, kw'^A^ir^) 

For the noncooperative solution corresponding to political weights a, the 
efficient pollution level is denoted by S(a; NNE) . Payoff function levels 
for consumers and producers at the joint maximum solution corresponding to 

the weights a are denoted by WTP(S(q; NNE); S°) , n^(a; NNE), tt'^Cq; NNE); 
these profit levels do not include taxes. 

For the status quo, the weight on the consumer is zero and producers 
are weighted equally. For this case, willingness to pay by consumers is 

zero and profit levels for producers are denoted by tt , tt . 

The noncooperative (NC) solution has the same water quality and food 
production as in the NNE case except that profits are reduced by taxes used 
to make individual choices correspond to the NNE solution. Taxes to be used 
for the noncooperative equilibrium are found from the first order conditions 
for the the NNE joint maximum problem. First order conditions for water use 
are : 

a ™ (5^ + 5\) a" + a ^ + a^ ^ - M A^ = 
c as u ^yu d ^„u 

3WTP ,^d, ^d ^ aTT^ id „ 

'^c "ai" ^^ ) ^ ^ '^d ,„d - ^ A = °- 

/i is the marginal cost associated with the water constraint; n will be zero 
if the water constraint is met, indicating that the water charge v is 
adequate for this purpose. Solving the first order conditions, the optimal 
tax is found: 

d-n^ _Ji ^u °'c aWTP ,,u ^ ^d, , .u ^ dn^ 

= — ^ A - — .„ (5 + 5 k) a - — 

.„u a a as a .,,u 

aw u u u aw 

a^ ^ _tt ^d ^ aWTP ^d ^d. 

aw^ «d ' -"d ^2 

the right hand sides of the above expressions (evaluated at the optimum 
technologies) are the optimal taxes on water use ( t^(Q) , t^(Q) ) for each 

producer as related to the weights a. 


Note that the optimal tax for an upslope producer is higher per unit of 
water used than a downslope producer for equal weights a , a , because the 

upslope producer causes externalities for both downslope producers and 
consumers. Water use is reduced by the tax compared to that for the status 


quo because the status quo solution satisfies - and — r 

for the 

same technology choices available in the NNE solution (see Figure 2) 

opt, tax 



Fig. 2. Water Use, Noncooperative Solution compared to the Status Quo. 

A tax on acres planted is also required to produce the NNE solution 
because externalities are caused by not only water use per acre but also by 
the acres planted by each type of producer. Optimal acres planted should 
satisfy the first order conditions: 


u a 

u \ 


dn^ ^ ^ ^ aWTP ^d yd 

..d a, a, as 
dA d d 


Optimal taxes t.(a) , i - u, d, on land use are again the right hand sides of 

the above conditions evaluated at the optimum technologies. A and A, 

denote land rental values ; if equal land areas are available to upslope and 
downslope producers and with equal weights, the upslope producer will pay a 
higher tax per unit land area than the downslope producer because of the 
externality effect on the downstream producer. Similarly to the effects of 
a tax on water use, less area will be planted than in the status quo case. 

Per acre profit levels for the noncooperative (NC) solution are 
obtained by subtracting taxes from NNE profits: 

n^(a; NC) - n^(a; NNE) - t'^(a) w"a" - t'f(Q) a'^ 

W X 

ff'^Ca; NC) - »r^(a; NNE) - t^(Q) w'^a'^ - t^(o) A^ 

Taxes could be placed on pollution directly, instead of on both water 
use and land use, to achieve the same joint maximum solution. However, 
since land and water use are more easily monitored than pollution outflows, 
enforcement and information costs would be lower with land and water taxes. 

The Cooperative Solution 
The joint welfare problems solved for the cooperative and 
noncooperative cases are similar except that only private technologies, 
chosen in response to taxes, are available in the noncooperative case, 
whereas in the cooperative case regional technolgies are also available and 
taxes are replaced by cost shares. In the cooperative case, drainage water 
quality is constrained to be at least as good as in the noncooperative case, 
but the cooperative case may also achieve a higher quality at lower joint 
cost because of economies of scale. 

Water quality of at least S(q; NNE) is to be achieved through the 


regional cooperative technologies (r ) combined with private water 
technologies. Regional water technology improvements include drainage 


systems, water treatment plants, and systems for storage and reuse of 

treated water. The regional joint regional cost of treating and reusing 

R R 
water is denoted by JC(S,W ;r ); note that these costs depend on the total 

volume of water (w ) and pollutant load (S). 

The optimization problem to be solved for the grand coalition is : 

JW(q; CS) - Max q^ WTP(S; S°)+ a^ [p^Y^ -0(7"^) - v'(r^)W^] a" 

,,u ,,d .u .d 
W , W ,A ,A , 

u d R 

T ,T , T 

+ a^IPfY^^ -c(r^) -v'(r^) w'^] a'^ - JC(S, W^; r^) 

s.t. a"<a" 


A^" + A^ W^ - W^ < W 

S < S(a; NNE) 

„u , u, u, ,.u .u 
S - 6 (r ) W A 

S - S'^ + S^ 

y" - Y^(w";r^) 

Y^ = Y^(W^, kW^A^; r^). 
Agreement to participate in the cooperative solution means that 
producers will be charged a share of the joint cost of the regional 
facility. Variable charges for water use will be reduced in the cooperative 
solution since tail water is reused on downslope lands and treated water is 
reused. Producers must also agree to use land, water, and irrigation 
technologies consistent with the joint welfare maximum. The pollution load 
achieved for the cooperative case will be denoted by S( a; CS) ; this will be 
no higher than S(a; NNE) since private technologies available are the same 

for both cooperative and noncooperative problems but regional technologies 
are also available here. Less costly private technologies may be used in 
the CS solution than in the NNE solution because the regional technologies 
will substitute for private technologies. 

Part of the cooperative game is to define how the costs of cooperative 
regional water improvements will be apportioned among consiimers and 
producers. Costs may be allocated according to the shares a. of the joint 

R R 
cost, that is, each producer and consumer will pay a share q.JC(S, W ; t ) 

so that joint costs are covered. 

The profit levels for each producer in the cooperative solution (CS) 

7r"(a; CS) - [P^y"" - cCr"") - v'(r^) w'^Ja'' - Q^JC(S, W^ ; r^) 

TT ^(a; CS) = [P^y'^ - c(t^) - v' (r^) w'^ja'^ - a^JC(S, W^ , r^) 

are evaluated at the optimal land and water use and technologies. 

More complicated cost allocation schemes have been proposed in game 
theory literature. Each has desirable features in terms of fairness, but 
one common feature is producing core solutions. The proposed cost shares 
procedure is simpler to compute and also can produce core solutions (see 
Appendix A). Its fairness property is consistency with political weights. 

Core Conditions 

Core conditions require that there be benefits of cooperation for the 
"grand coalition" (here all producers and consumers) and also that no 
subcoalition could improve on their payoffs in the grand coalition. 

For producers, there will be a gain from cooperation in the grand 
coalition if profit for the cooperative solution exceeds that for the 
noncooperative case : 

7r^(Q; CS) - 7r^(a; NC) > 0. 


The technology choice set for the noncooperative problem is contained in 
that for the cooperative problem. Therefore, an increase in profits 
compared to the noncooperative solution will occur if the joint cost share 
is less than the tax cost in the noncooperative solution, private 
technologies in the cooperative case are less expensive, and output is not 
reduced. For the stronger acceptability criterion (positive benefits of 
cooperation compared to the status quo) , the following additional 
requirement should hold: 

tt^Cq; CS) - tt^ > 0. 
o — 

Whether or not either condition holds will depend on political weights and 

the nature of the technologies available. 

For consumers, water quality is at least as good in the cooperative 

solution as in the noncooperative. The cooperative solution with 

proportional cost shares will give the result for consumers that: 

WTP(S(a; CS) ; S°) - q JC > 0. 

However, since consumers do not have to pay anything in the noncooperative 
case, for consumers to be better off in the cooperative solution compared to 
the noncooperative case, the following condition is also required: 

WTP(S(a; CS) ; S°) - WTP(S(a; NNE) ; S°) > a JC. 

That is, the improvement in water quality for the cooperative solution must 
be sufficient to offset the cost share paid by consumers. Again, whether 
this holds will depend on relative weights and technologies. Consumers are 
better off in the noncooperative case compared to the status quo because 
water quality is better. 

To determine cases when the cooperative solution will be in the core, 
empirical work should be undertaken. Such work would determine the size of 
the payoff gain for cooperative as compared to noncooperative solutions. 


This paper has developed a series of models which can be used to design 
incentives on a regional basis so that producers would voluntarily adopt 
improved technologies and agree to participate in a regional water 
management system. These incentives include taxes for a noncooperative 
solution and cost shares for an alternative cooperative solution. Both 
types of solutions are defined to be consistent with political weights and 
economic efficiency. 

Taxes paid on water use per acre and acres planted are used as a 
noncooperative "threat point". Higher tax levels should be charged to 
upslope producers because of greater externality costs associated with 
production. In the alternative cooperative case, there are incentives for 
cooperation if cost shares are less than tax payments. 

Empirical work should be undertaken to determine when the cooperative 
solution satisfies core conditions given the nature of alternative 
technologies and political weights. A larger positive difference in gains 
between cooperative and noncooperative solutions would more likely lead to 
voluntary acceptance of a cooperative solution. To determine when 
cooperative solutions would voluntarily be chosen over noncooperative 
solutions, experimental games (Ostrom and Gardner, 1990) could also be 



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Appendix A 
Cost Allocation for a Regional Cooperative Game 
Cost allocation for public goods has been modelled as a cooperative 
game. Denote the joint cost for water technologies for a coalition S of 
players by JC(S) . Here, the grand coalition is the set of all players 
{u,d,c) where u,d,c, denote respectively upstream and downstream producers 
and consumers. Here, technologies associated with the joint cost function 
must achieve at least the drainage water quality associated with the NNE 
threat point. If the solution for the game is based on the grand coalition, 
the cost shares (x.) for each player must satisfy 

X + X , + X - JC(u,d,c) 
u d c 

to cover total cost of a cooperative arrangement. Subcoalitions of players 

may also form to satisy the same environmental constraints if they may do so 

more cheaply than in the grand coaltion. 

In a cooperative cost allocation problem, the core respresents the set 

of points which are rational, in that each player is better off in the 

cooperative game than acting alone, and there are also benefits of 

cooperation for each subcoalition of players. Therefore, cost allocations 

in the core of the cooperative game must also satisfy 

X + X , < JC(u,d) 
u d ~ 

X + X < JC(u,c) 
u c ~ 

X , + X < JC(d,c) 
d c 

Figure 3 illustrates the core for three players. 

The Shapley value has been proposed as a solution in cooperative cost 

allocation games because it satisfies a fairness axiom as well as covering 

joint costs (it also satisfies a technical requirement - linearity in terras 

of different cost allocation problems). The Shapley value will be in the 

core provided that there are economies of scale. For the Shapley value. 



Figure 3. Illustration of the Core with Three Players. 

each vertex of the core (see Figure 3) is weighted equally and the Shapley 
value is the center of gravity of the core. 

Alternatively, cost allocation can also be based on the generalized 
Shapley value (Loehman and Whinston, 1976; Loehman et al, 1979). For the 
generalized Shapley value, vertices of the core can have different weights; 
the resulting value is again the center of gravity of the core. Alternative 
points in the core can be obtained simply by varying the weights on the 
vertices. Generalized Shapley values have also been discussed by Kalai and 
Samet (1987) based on probability weights. 

The form of the generalized Shapley value is defined in terms of the 
weighted incremental costs [JC(S) - JC(S-i)] caused by the addition of a 
player to a coalition: 

X. ■== S 7-(S) [JC(S) - JC(S - i)] 

where S denotes a subcoalition containing player i and JC(S) is the cost for 
the subcoalition; the weights 7.(S) are determined by the weights on the 

vertices of the core such that the costs shares add up to the total cost. 


For the generalized Shapley value applied to this problem with the 
three players (u, d, c), cost shares for each participant are defined by: 
'^u " ^u^""^ -^^^^^ ^ 7^((udc)) [JC({u,d,c)) - JC((d.c))] 

+ 7y({uc)) [JC({u,c}) - JC(c)] + 7^({ud)) [JC({u,d)) - JC(d)]; 
^d ' "^d^^^ -^^^^^ "^ T^dudc)) [JC((u,d,c)) - JC((u,c))] 

+ 7^((ud)) [JC({u,d)) - JC(u)] + 7^({cd)) [JC((c,d)) - JC(c)l 
x^ = 7^(c) JC(c) + 7^({udc)) [JC((u,d,c)) - JC((u,d))] 

+ 7j.(luc)) [JC({u,c}) - JC(u)] + 7^({dc)) [JC((d,c)) - JC(d)]; 
for weights 7^(S) defined in terms of vertex weights as follows: 

7^(ud) - ^2 7jj(ud) - ^^ 7^(dc) = u^ 

7^(uc) = u^ 7^(dc) - ^^ 7^(uc) - i/g 

7^(udc) =1/^ + 1/^ 7^(udc) - 1^5 + 1^5 7^(udc) = ''j^ + ^2 
An alternative representation of core points can be based on weights a. 
on the points of the triangle in Figure 3, where a. represent the political 

weights for each player. To give the center of gravity of the triangle, 
costs shares must satisfy 

x^ - a^JC((u,d,c)). 

The center of gravity of the triangle and the center of gravity of the core 
will correspond for a relationship between the weights 7. on the vertices of 

the core and the weights q. on the vertices of the triangle. Therefore, 

cost allocation based on weighted shares will also represent a core 
allocation when there are economies of scale. 

This correspondence between weights is given by 

a, a 

d c 


1 Q +Q, a +Q:, +Q 

u d u d c 

a a 

u c 

2 a + a, a + a, + a 
u d u d c 

a a 

c u 

3 a, +Q a + a. + a 
a c u d c 

a, a 
d u 



a + 






^ ^d 

+ a 


a + 




+ '^d 

+ a 



6 Q +Q Qt +a.+Q' 
u c u d c 



Appendix B 

Empirical Specification of Consumer Benefits 

Consumer Health Benefits Model 

Willingness to pay in terms of health (WTPH) to reduce S below current 
level S , to be multiplied by the number of residents, is estimated as 

follows . 

1) Willingnes to pay in terms of mild symptoms, per person per year: 

WTP = $.05 (700-SC) per person. 

SC = /ig/day of selemium consumption: 700 ^ig => mild symptoms; 

250 /ig => no symptoms. 
Max WTP to avoid mild symptoms: WTP - $22,50, SC < 250. 

2) Selenium cone, in body related to importance of consumption source. 

From Chinese studies: 

SC = 1.525 SF + 9.13 SW 

SF = selenium from food consumption, ^g/day 
SW - water concentration, ppb 

SC = 700 Mg/day 

SF = 250 ^ig/day (1/2 of Japan is consumption) 

SW = 35 ppb. 

3) Selenium cone, as related to shallow water table: 

S = selenium concentration in shallow water table. 
SF = 2.5 S 
SW - .35 S 
Currently: S = 100 ppb. 

4) Combining the above relationships: 

WTPH - .05 (700 - 7S) - .35 (100 - S) 
[Note: S = 100 => WTP = 0] 


Fishing Benefit 

Fishing benefit (WTPF) will be measured by the change in number of 
fish caught per fishing person per year, to be multiplied by the value per 
fish. Birdwatching benefit will be similarly defined. 

1) Reproductive rate loss (RL) : 

RL - 7 SCF 

SCF - whole body cone, of selenium in fish 

Note: RL - .3 when SCF - 100 ^g -> 7 - .003. 

2) Selenium cone, in fish related to water quality: 

SCF - /9 SW 

SCF - 100 /ig/g at San Luis Drain 

SW = 70 ppb " " " (twice the average in San Joaquin) 

"> /3 - 1.43 

3) Water concentration as related to shallow water table: 

SW - .35 S 

4) Combining, the reproductive loss as related to the shallow water table: 

RL = (.003)(1.43)(.35)S 
= .0015 S 
Implications: Reduce S -> RG - .0015 (100 - S) , i.e. the reproductive 
gain per fish is 15% if S is reduced from 100 to zero. 

5) Valuing the reproductive gain: 

Fishing day: catch 2 fish; value of a fishing day is worth $20. 
Implication: each fish is worth $10. 

WTPF = ($10)(.0015)(100 - S) - .015(100-S) 

6) Multiply 5) by the number of fishing days per year times the number of 

fishermen in the area.